I made a list of clinics in central Tokyo with staff who can help you in English and will take National Health Insurance, so you won’t have to pay an arm and a leg. Just click the hyperlink to see it. I also placed it on the menu of the main page for easy access.
There are other sites online with much more comprehensive lists, but the ones I put here will be places I (or a friend) have personally been to and have confirmed to be English-capable and will take Japanese insurance.
The list is still quite short, but will add to it as I find more.